  (Solution) IEOR E4404 Simulation, Spring 2015 April 16, 2015 Assignment 5 Due Date: April 30, 2015 Problem 1. Of The Form Suppose That We Wish To Estimate =... | Snapessays.com

(Solution) IEOR E4404 Simulation, Spring 2015 April 16, 2015 Assignment 5 Due date: April 30, 2015 Problem 1. of the form Suppose that we wish to estimate =...

April 16, 2015

Assignment 5

Due date: April 30, 2015

Problem 1.

Suppose that we wish to estimate

?

=

E

[

X

], and that we design a control variate estimator

of the form

¯

Z

n

(

?

*

) =

1

n

n

X

i

=1

Z

i

(

?

*

)

,

where

Z

i

(

?

*

) =

X

i

+

?

*

(

Y

i

-

?

),

X

i

are i.i.d copies of

X

,

Y

i

are i.i.d copies of the control random

variable

Y

, and

E

[

Y

] =

?

is known exactly. Suppose that

?

*

is selected optimally, and you know its

value. Furthermore, assume that it takes 1 unit of computer time to obtain each

X

i

, but

K

units of

computer time to obtain each

Z

i

(

?

*

). What is the range of values of

?

xy

, where

?

xy

is the correlation

between

X

and

Y

, as a function of

K

, for us to justify the use of this control variate method? Show

Problem 2.

The control variate method describe in class makes use of one control random variable.

Sometimes it is also possible to use two control random variables. Suppose that we are interested in

estimating

?

=

E

[

X

], and suppose that random variables

Y

and

Z

are both correlated with

X

. Consider

the random variable

W

(

?

1

,?

2

) =

X

+

?

1

(

Y

-

?

1

) +

?

2

(

Z

-

?

2

)

,

where

?

1

=

E

[

Y

] and

?

2

=

E

[

Z

].

(a) Show that

E

[

W

(

?

1

,?

2

)] =

?

for any

?

1

and

?

2

.

(b) Find expressions for

?

*

1

and

?

*

2

which minimize the variance of

W

(

?

1

,?

2

).

(c) Suppose that

X

and

Y

are positively correlated, and that

X

and

Z

are positively correlated as

well. Provide an example where

?

1

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